Yukio Hirashita
What are the prices of random variables? In this paper, we define the least-squares prices of coin-flipping games, which are proved to be minimal, positive linear, and arbitrage-free. These prices depend both on a set of games that are available for investing simultaneously and on a risk-free interest rate. In addition, we show a case where the mean-variance portfolio theory is inappropriate.
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This paper is intended to help the readers to understand the article: Y. Hirashita, Least-Squares Prices of Games, Preprint (arXiv:math.OC/0703079).
The price of anarchy, originally introduced to quantify the inefficiency of selfish behavior in routing games, is extended to mean field games. The price of anarchy is defined as the ratio of a worst case social cost computed for a mean field game equilibrium to the optimal social cost as computed by a central planner. We illustrate properties of such a price of anarchy on linear quadratic extended mean field games, for which explicit computations are possible. Various asymptotic behaviors of the price of anarchy are proved for limiting behaviors of the coefficients in the model and numerics a…
In this paper, we introduce and study a class of games called price-coupling games that arise in many scenarios, especially in the electricity industry. In a price-coupling game, there is a part of the objective function of a player which has an identical form for all players and there is coupling in the cost functions of players through a price which is determined uniformly for all players by an independent entity called the \textit{price-determining player} (e.g. independent system operator (ISO) in an electricity market). This price appears in the objective function only in the part which i…
The petrol station game: the regional average price. A mean field analysis
本文提出一个平均场博弈模型，分析意大利政府强制加油站显示区域平均油价政策下的价格演化，研究平均场均衡的存在性与演化，并用特伦托省的真实数据验证模型。
We consider nonatomic network games with one source and one destination. We examine the asymptotic behavior of the price of anarchy as the inflow increases. In accordance with some empirical observations, we show that, under suitable conditions, the price of anarchy is asymptotic to one. We show with some counterexamples that this is not always the case. The counterexamples occur in very simple parallel graphs.
In this paper we study Mean Field Game systems under density constraints as optimality conditions of two optimization problems in duality. A weak solution of the system contains an extra term, an additional price imposed on the saturated zones. We show that this price corresponds to the pressure field from the models of incompressible Euler's equations {à} la Brenier. By this observation we manage to obtain a minimal regularity, which allows to write optimality conditions at the level of single agent trajectories and to define a weak notion of Nash equilibrium for our model.
We consider the problem of optimal charging of plug-in electric vehicles (PEVs). We treat this problem as a multi-agent game, where vehicles/agents are heterogeneous since they are subject to possibly different constraints. Under the assumption that electricity price is affine in total demand, we show that, for any finite number of heterogeneous agents, the PEV charging control game admits a unique Nash equilibrium, which is the optimizer of an auxiliary minimization program. We are also able to quantify the asymptotic behaviour of the price of anarchy for this class of games. More precisely, …
We generalize the notions of user equilibrium and system optimum to non-atomic congestion games with stochastic demands. We establish upper bounds on the price of anarchy for three different settings of link cost functions and demand distributions, namely, (a) affine cost functions and general distributions, (b) polynomial cost functions and general positive-valued distributions, and (c) polynomial cost functions and the normal distributions. All the upper bounds are tight in some special cases, including the case of deterministic demands.
Prices of Anarchy, Information, and Cooperation in Differential Games
本文将无政府代价（PoA）概念扩展到一般微分博弈框架，并引入信息代价（PoI）和合作代价来量化不同信息结构和利他行为下的效率损失或收益，针对标量线性二次微分博弈给出了开环和闭环反馈信息结构下的显式界。
Branch-and-price with novel cuts, and a new Stackelberg Security Game
该研究针对关键基础设施防护的Stackelberg安全博弈问题，提出新型有效不等式求解强Stackelberg均衡，结合分支定价法验证了方法优于现有标准公式，并分析了参数对性能的影响。
Markov Games with Decoupled Dynamics: Price of Anarchy and Sample Complexity
本文研究动态解耦但奖励可耦合的马尔可夫博弈，引入平滑性概念界定无政府代价，并为马尔可夫势博弈提出可证明收敛至纳什均衡的分布式学习算法MA-SPI及样本复杂度。
A primal-dual price-optimization method for computing equilibrium prices in mean-field games models
本文提出一种拉格朗日原始-对偶梯度算法，用于计算平均场博弈价格形成模型中的均衡价格，利用自动微分技术实现高效灵活的模块化计算，可扩展至高维复杂设置且无需手动推导伴随方程。
The price of anarchy has become a standard measure of the efficiency of equilibria in games. Most of the literature in this area has focused on establishing worst-case bounds for specific classes of games, such as routing games or more general congestion games. Recently, the price of anarchy in routing games has been studied as a function of the traffic demand, providing asymptotic results in light and heavy traffic. The aim of this paper is to study the price of anarchy in nonatomic routing games in the intermediate region of the demand. To achieve this goal, we begin by establishing some smo…
We develop a model for the industry dynamics in the electricity market, based on mean-field games of optimal stopping. In our model, there are two types of agents: the renewable producers and the conventional producers. The renewable producers choose the optimal moment to build new renewable plants, and the conventional producers choose the optimal moment to exit the market. The agents interact through the market price, determined by matching the aggregate supply of the two types of producers with an exogenous demand function. Using a relaxed formulation of optimal stopping mean-field games, w…
The Calculation and Simulation of the Price of Anarchy for Network Formation Games
本文通过整数规划计算网络形成博弈中的无政府状态价格，并模拟网络形成过程，发现模拟的无政府状态价格往往较低。
This paper deals with a stochastic order-driven market model with waiting costs, for order books with heterogenous traders. Offer and demand of liquidity drives price formation and traders anticipate future evolutions of the order book. The natural framework we use is mean field game theory, a class of stochastic differential games with a continuum of anonymous players. Several sources of heterogeneity are considered including the mean size of orders. Thus we are able to consider the coexistence of Institutional Investors and High Frequency Traders (HFT). We provide both analytical solutions a…
For a game with positive profit, the optimal proportion of investment required to continue investing without borrowing is uniquely determined by an integral equation for each price. For a game with parallel translated profit, the ratio of the optimal proportion of investment to its price has some invariance properties. The optimal price of a game with parallel translated profit converges to its expectation divided by e to the riskless interest rate for a certain period.
We consider the capacitated selfish replication (CSR) game with binary preferences, over general undirected networks. We first show that such games have an associated ordinary potential function, and hence always admit a pure-strategy Nash equilibrium (NE). Further, when the minimum degree of the network and the number of resources are of the same order, there exists an exact polynomial time algorithm which can find a NE. Following this, we study the price of anarchy of such games, and show that it is bounded above by 3; we further provide some instances for which the price of anarchy is at le…
For a game with positive expectation and some negative profit, a unique price exists, at which the optimal proportion of investment reaches its maximum. For a game with parallel translated profit, the ratio of this price to its expectation tends to converge toward less than or equal to 1/2 if its expectation converges to 0. In this paper, we will investigate such properties by using the integral representations of a complete Bernstein function and establish several Abelian and Tauberian theorems.
We study a static game played by a finite number of agents, in which agents are assigned independent and identically distributed random types and each agent minimizes its objective function by choosing from a set of admissible actions that depends on its type. The game is anonymous in the sense that the objective function of each agent depends on the actions of other agents only through the empirical distribution of their type-action pairs. We study the asymptotic behavior of Nash equilibria, as the number of agents tends to infinity, first by deriving laws of large numbers characterizes almos…
We study a new kind of non-zero-sum stochastic differential game with mixed impulse/switching controls, motivated by strategic competition in commodity markets. A representative upstream firm produces a commodity that is used by a representative downstream firm to produce a final consumption good. Both firms can influence the price of the commodity. By shutting down or increasing generation capacities, the upstream firm influences the price with impulses. By switching (or not) to a substitute, the downstream firm influences the drift of the commodity price process. We study the resulting impul…
On the Efficiency of Nash Equilibria in Aggregative Charging Games
本文研究了在聚合充电博弈中纳什均衡的效率问题，通过分析纳什均衡与社会最优解之间的价格错配，得出了在不同价格函数情况下纳什均衡效率趋近于最优的结论，并提供了有限人口下的效率界限。
Nash equilibrium in a singular stochastic game between two renewable power producers with price impact
本文研究了两个可再生能源生产者在电力价格受其累计安装影响下的奇异随机博弈，通过静态和动态分析，探讨了纳什均衡的形成及自由边界问题。
Phase Transitions of the Price-of-Anarchy Function in Multi-Commodity Routing Games
本文研究多商品路由博弈中无政府代价随需求变化的相变，证明比例需求下断点处单侧导数与较小均衡支撑集关联，并给出非比例需求的反例。
Regularity of the Price of Anarchy in Routing Games over the Multivariate Demand Space
该研究分析多变量需求空间下路由博弈的无政府状态价格（PoA）的正则性，证明其在正则需求区域连续可微，不可微集测度为零，仿射成本下PoA最大值在单元边界取得，并在Sioux Falls网络验证理论。
Renewable resources are starting to constitute a growing portion of the total generation mix of the power system. A key difference between renewables and traditional generators is that many renewable resources are managed by individuals, especially in the distribution system. In this paper, we study the capacity investment and pricing problem, where multiple renewable producers compete in a decentralized market. It is known that most deterministic capacity games tend to result in very inefficient equilibria, even when there are a large number of similar players. In contrast, we show that due t…
Stochastic Approximation for Estimating the Price of Stability in Stochastic Nash Games
本文针对随机纳什博弈中稳定价格（PoS）难以高效计算的问题，提出基于随机逼近的求解方案，并给出近似误差界，为网络化系统效率评估提供了新方法。
We propose a model where a producer and a consumer can affect the price dynamics of some commodity controlling drift and volatility of, respectively, the production rate and the consumption rate. We assume that the producer has a short position in a forward contract on λunits of the underlying at a fixed price F, while the consumer has the corresponding long position. Moreover, both players are risk-averse with respect to their financial position and their risk aversions are modelled through an integrated-variance penalization. We study the impact of risk aversion on the interaction between th…
Game Design and Analysis for Price based Demand Response: An Aggregate Game Approach
本文提出了一种聚合游戏方法，用于智能电网中能源消费控制的建模与分析。通过估计聚合负荷，提出了纳什均衡寻求策略，并利用奇异摄动分析和李雅普诺夫稳定性分析，证明了局部收敛性。
A Stackelberg Game Approach to Control the Overall Load Consumption of a Residential Neighborhood
提出协调智能体与家庭间的 Stackelberg 博弈及基于 KKT 条件、隐函数定理和梯度下降的分布式定价框架，实现住宅小区负荷整形并优于商业集中式求解。
In this paper we formulate the now classical problem of optimal liquidation (or optimal trading) inside a Mean Field Game (MFG). This is a noticeable change since usually mathematical frameworks focus on one large trader in front of a "background noise" (or "mean field"). In standard frameworks, the interactions between the large trader and the price are a temporary and a permanent market impact terms, the latter influencing the public price. In this paper the trader faces the uncertainty of fair price changes too but not only. He has to deal with price changes generated by other similar marke…
We consider a game where a finite number of retailers choose a location, given that their potential consumers are distributed on a network. Retailers do not compete on price but only on location, therefore each consumer shops at the closest store. We show that when the number of retailers is large enough, the game admits a pure Nash equilibrium and we construct it. We then compare the equilibrium cost borne by the consumers with the cost that could be achieved if the retailers followed the dictate of a benevolent planner. We perform this comparison in term of the price of anarchy, i.e., the ra…
Subsidizing a New Technology: An Impulse Stackelberg Game Approach
针对零排放车辆补贴问题，本文构建含企业连续定价与政府离散补贴干预的脉冲 Stackelberg 博弈，并用验证定理刻画 Feedback Stackelberg 均衡。
Price-Coordinated Mean Field Games with State Augmentation for Decentralized Battery Charging
本文利用均场博弈解决大规模电池存储代理的去中心化协调充电问题，通过状态增强将充电功率作为状态变量，其变化率作为控制输入，推导出非线性耦合的前向后向微分方程，并证明了均衡的存在性和唯一性。
We study optimal behavior of energy producers under a CO_2 emission abatement program. We focus on a two-player discrete-time model where each producer is sequentially optimizing her emission and production schedules. The game-theoretic aspect is captured through a reduced-form price-impact model for the CO_2 allowance price. Such duopolistic competition results in a new type of a non-zero-sum stochastic switching game on finite horizon. Existence of game Nash equilibria is established through generalization to randomized switching strategies. No uniqueness is possible and we therefore conside…
Paradoxes of Game Theoretic Equilibria and Price of Anarchy
研究博弈论均衡与无政府价格的悖论，指出将多智能体学习简化为静态均衡和黑箱后悔分析存在问题，通过分析内部纳什均衡、拥堵博弈等情况揭示相关不足，并研究非原子极限下的均衡变化，强调需重新评估基于动态的最坏情况均衡框架。
A market with asymmetric information can be viewed as a repeated exchange game between the informed sector and the uninformed one. In a market with risk-neutral agents, De Meyer [2010] proves that the price process should be a particular kind of Brownian martingale called CMMV. This type of dynamics is due to the strategic use of their private information by the informed agents. In the current paper, we consider the more realistic case where agents on the market are risk-averse. This case is much more complex to analyze as it leads to a non-zero-sum game. Our main result is that the price proc…
Strategic Index Reconstitution: Differential Games, Closed-Loop Equilibria and Mean-Field Dynamics
本文通过连续时间多资产微分博弈建模指数调整中的策略性交易，构造子博弈完美纳什均衡与均值场均衡，揭示竞争和冲击衰减对价格偏移与执行成本的影响。
Deriving the Pure Price of Anarchy for Networked Resource Allocation Games
本研究提出线性规划方法，为任意信息网络和系统目标推导最优纯无政府状态价格，并证明超模目标下完全禁止通信的效用设计最优，子模目标下设计具鲁棒性。
Solar Renewable Energy Certificate (SREC) markets are a market-based system that incentivizes solar energy generation. A regulatory body imposes a lower bound on the amount of energy each regulated firm must generate via solar means, providing them with a tradeable certificate for each MWh generated. Firms seek to navigate the market optimally by modulating their SREC generation and trading rates. As such, the SREC market can be viewed as a stochastic game, where agents interact through the SREC price. We study this stochastic game by solving the mean-field game (MFG) limit with sub-population…
A Constrained Evolutionary Gaussian Multiple Access Channel Game
针对具有连续变量动作和耦合速率约束的演化多址信道博弈，刻画了纯纳什均衡的帕累托最优性和强均衡性质，并利用规范化均衡和演化稳定策略选择特定均衡解，分析了布朗-冯·诺伊曼-纳什动力学和复制动力学下的长期行为。
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